A024792 Number of 8's in all partitions of n.

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 23, 31, 44, 59, 82, 108, 146, 191, 254, 328, 429, 549, 709, 900, 1148, 1446, 1829, 2286, 2865, 3559, 4427, 5465, 6752, 8288, 10178, 12429, 15175, 18442, 22404, 27102, 32767, 39473, 47516, 57012, 68349, 81703, 97579, 116236

Offset

1,10

Comments

The sums of eight successive terms give A000070. - Omar E. Pol, Jul 12 2012

a(n) is also the difference between the sum of 8th largest and the sum of 9th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.

Formula

a(n) = A181187(n,8) - A181187(n,9). - Omar E. Pol, Oct 25 2012

From Peter Bala, Dec 26 2013: (Start)

a(n+8) - a(n) = A000041(n). a(n) + a(n+4) = A024788(n).

a(n) + a(n+2) + a(n+4) + a(n+6) = A024786(n).

O.g.f.: x^8/(1 - x^8) * product {k >= 1} 1/(1 - x^k) = x^8 + x^9 + 2*x^10 + 3*x^11 + ....

Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*Pi*sqrt(2*n)) * (1 - 97*Pi/(24*sqrt(6*n)) + (97/48 + 6337*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016

Maple

b:= proc(n, i) option remember; local g;
      if n=0 or i=1 then [1, 0]
    else g:= `if`(i>n, [0$2], b(n-i, i));
         b(n, i-1) +g +[0, `if`(i=8, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

Mathematica

Table[ Count[ Flatten[ IntegerPartitions[n]], 8], {n, 1, 53} ]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 8, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 09 2015, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A066633, A024786, A024787, A024788, A024789, A024790, A024791, A024793, A024794.

Sequence in context: A321142 A336106 A366845 * A280661 A340659 A055771

Adjacent sequences: A024789 A024790 A024791 * A024793 A024794 A024795

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved